2024-06-18
Allow for staggered adoption
Reduce interpolation
Account for exposed series outside of the convex hull
Incorporate control series on different outcomes/scales from treated series
We saw in the previous example that multiple states initiated lotteries at different times.
Plot of percent fully vaccinated rates by U.S. state, Jan.-Sept. 2021
The standard SCM had weights \(w_i\) that minimized:
\[ \sum_{k=1}^K v_k \left( X_{1k} - \left(\sum_{i=1}^n X_{0ik} w_i \right) \right)^2, \]
for covariates \(k=1,\ldots,K\).
This can be extended with a penalty term \(\xi\) on a function \(f(w_i)\) to minimize:
\[ \sum_{k=1}^K v_k \left( X_{1k} - \left(\sum_{i=1}^n X_{0ik} w_i \right) \right)^2 + \xi \sum_{i=1}^n f(w_i). \]
The penalty can reduce interpolation (force closer matches to specific units) or reduce discrepancy from an outcome model.
Idea
De-mean the pre-treatment data, fit SC to the de-meaned observations, and apply the weights to both the post-treatment time trends and levels.
Incorporates idea of diff-in-diff of focusing on matching trends instead of levels.
Warning
Matching pre-treatment trends may not lead to stable weights going forward.
Different interpretation of weights.
Idea
Incorporate unit weighting of SC with unit fixed effects of DID and time weighting. “Localized” TWFE model.
Idea
De-bias SC estimate using an outcome model for the time series.
If pre-treatment fit is good, discrepancy will be small and adjustment will have little effect
Allows for level shift by capturing a consistent discrepancy
Can still express as weights of control units, but negative weights now allowed; penalizes discrepancy from SC weights
Options:
Fit separate SCM for each unit
Fit SCM on average of treated units
Partially pooled SCM: mix of both
With intercepts, similar trade-offs to weighted DID approaches
Advantages:
Disadvantages:
Idea: “Interactive Fixed Effects”
Use control unit data to estimate unit fixed effects and time-varying coefficients. Use these coefficients to estimate treated unit fixed effects and then counterfactuals.
Allows extrapolation and relies on modeling of time series. But can achieve better fit, especially with shift. Allows multiple treated units to be estimated quickly.
Idea
Combine three information sources in state-time model:
Pooling across units enables better fit, but loses unit-specific information
Extrapolation allows better pre-treatment fit, but may over-fit or rely on additional assumptions
Reducing interpolation is crucial for some settings (non-linear outcomes)
Advanced methods allow use of more control series: reduces variance but may introduce bias
A key benefit of SC is its interpretability
This is somewhat lost in more advanced approaches
The interpretability is tied to justifying the assumptions as well